A treatise on the theory of Bessel functions, by G. N. Watson.

318

THEORY OF BESSEL FUNCTIONS

[CHAP. X

It follows that the expansion (1) is valid when (a, 8/) lies in any of the
domains 1, 2, 3.
Next, we have to consider the asymptotic expansion when (a, /3) does not
lie in any of these domains. To effect our purpose we have to determine the
destinations of the branch of the curve D (u; v) = 0 which passes through the
origin.
Consider first the case in which a is positive and f is acute. The function
'D (a, v) has maxima at v = (2n + 1) vr- 3 and minima at v =(2n + 1) Tr + 3,
each minimum being greater than the preceding; and since p (a, /3 - r) is now
positive, it follows that P (a, v) is positive when v is greater than - V.
Hence the curve cannot cross the line u = a above the point at which
v = - r, and similarly it cannot cross the line u = - a below the point at which
v = 7. The branch which goes downwards at the origin is therefore confined
to the strip - a < u < a until it gets below the line v = - 2Kr + r - /3, where
K is the smallest integer for which
1 -a tanha+ {(2K+ 1)7r-/} cot3 > 0.
The curve cannot cross the line v =- (2K + 1) r +,, and so it crosses the
line u = a and goes off to infinity in the direction of the line v = - 2Kr.
Hence, if a is positive and f is acute, we get
1 f0-2K ri 1 - (2m)! a
(2) e-vt-zsinh tdt~- 1 (2 m
Vr 7r 0=O Z2M+
while, if a is negative and / is acute, we get
~I(3) 1 e2K-i I - (2m)!am
(3) e-vt-zsinhtdt,_ — 1I E.21m
7rJo 7r m=0O z
By combining these results with those obtained in ~ 8'61, we obtain the
asymptotic expansions for the domains 6 a and 7 a.
If, however, /i is obtuse and a is positive, the branch which goes below the
axis of u at the origin cannot cross the line u = a below (a, wr - /) and it does not
cross the u-axis again, so it must go to - oc along the line v =- (2L + 1) 7r,
where L is the smallest integer for which
1 -a tanh a - {(2L + 1) 7r + /3 cot/3 > 0.
Hence, if a is positive and /3 is obtuse, we get
(4) 1| ) e-vt-z sinht, 1 (2n)! a
7J 0 v m=O z
while, if a is negative and / is obtuse, we get
(51 + (2+l)Ti+ 1 E (2NI.)! a,,
(5) -1 evt-zsinhtdt — I (,L+
rrju y T^=